The violation of a uniqueness theorem and an invariant in the application of Poincaré--Perron theorem to Heun's equation (1605.08960v4)

Abstract: The domain of convergence of a Heun function obtained through the Poincar\'{e}--Perron (P--P) theorem is not absolute convergence but conditional one [2]. We show that a uniqueness theorem is not available if we apply the P--P theorem into the Heun's equation. We verify that the uniqueness theorem is only applicable when a local Heun function is absolutely convergent.